A gear reduction unit uses the countershaft shown in the figure. Gear

A receives power from another gear with the transmitted force $F_A$

applied at the $20$ pressure angle as shown. The power is transmitted

through the shaft and delivered through gear B through a transmitted

force FB at the pressure angle shown.

a$)$ Determine the force $F_n,$ assuming the shaft is running at a constant

speed.

b$)$ Find the magnitudes of the bearing reaction forces, assuming the

bearings act as simple supports.

c$)$ Draw shear$-$force and bending$-$moment diagrams for the shaft. If

needed, make one set for the horizontal plane and another set for the

vertical plane.

d$)$ Determine the minimum safe diameter of the shaft considering a

factor of safety of $3\mathrm{.}5$ and yield strength of $60k.$ Use distortion

energy theory

e$)$ Determine the minimum safe diameter of the shaft considering a

factor of safety of $3\mathrm{.}5$ and yield strength of $60k.$ Use maximum

shear stress theory

f$)$ Based on the result of distortion energy theory and maximum shear

stress theory, choose the safe diameter on Table A$-17.$

g$)$ Determine the fatigue factor of safety using modified$-$Goodman

criteria

h$)$ Determine the fatigue factor of safety using Gerber criteria

i$)$ Determine the fatigue factor of safety using ASME$-$Elliptic criteria